🤖 AI Summary
This study proposes a rigorous, optimization-free definition of sustainability from a systems dynamics perspective, characterizing the sustainability of the coupled Earth–human–production system as a geometric property: namely, that system trajectories almost surely avoid collapse boundaries. By constructing a stochastic differential equation model with multiplicative noise and employing Feller boundary classification alongside stochastic Lyapunov methods, the work integrates a reflexive social evaluation functional and endogenous biodiversity. It identifies the sign of net cross-subsystem flows as a critical order parameter governing phase transitions. The analysis demonstrates that when net inter-system flows are negative in a neighborhood of zero, collapse occurs almost surely. Sustainable development requires the evaluation functional to be locally increasing and positively aligned with survival probability, yielding the first path-dependent welfare comparison mechanism that does not rely on optimization assumptions.
📝 Abstract
We construct a stochastic dynamical systems theory in which sustainability is a structural boundary property of a fully coupled Earth--Human--Production system. Each subsystem is modelled as a vector-valued process governed by stochastic differential equations with multiplicative noise and absolute bidirectional cross-subsystem flows. Biodiversity is endogenous, and societal evaluation is represented by a reflexive functional whose weights depend on evolving human capabilities. Sustainability, development, and sustainable development are defined as trajectory properties. Sustainability corresponds to boundary non-attainment with positive or unit probability; development corresponds to local ascent in the evaluation functional; sustainable development requires directional alignment under strictly positive survival probability. No optimisation problem is imposed. Necessary and sufficient conditions are derived using Feller boundary classification and stochastic Lyapunov methods. A central result identifies the sign of the net absolute cross-subsystem flow on each component as a phase-transition parameter: if negative near zero, the boundary is of exit type and almost-sure persistence is structurally impossible, independent of intrinsic regeneration, capability accumulation, or productivity parameters. Because flows are absolute, any perturbation diffuses through the entire coupled system without requiring correlated exogenous shocks. The reflexive evaluation structure generically induces non-transitive development relations, providing a formal mechanism for path-dependent welfare comparisons. Sustainability emerges as a geometric property of boundary structure and vector-field alignment, not as a corollary of intertemporal optimality.